- #1
maverick280857
- 1,789
- 5
Hi
Here's a question...
Let a, b, c be three non-zero real numbers such that
a + b + c = abc
Prove that at least one of these three numbers (a, b or c) is less than or equal to the square root of 3.
Can you prove this without trigonometry? The trigonometric solution follows...
Solution (using Trigonometry)
Let a = tan(A), b = tan(B), c = tan(C) (for some nonzero angles A, B, C which are real) so that the given constraint becomes
tan(A) + tan(B) + tan(C) = tan(A)tan(B)tan(C)
which can be true iff A + B + C = n*pie (n is an integer)
If n = 1, then A, B, C are the angles of a triangle (as the constraint is true for angles of a triangle). The result follows by considering cases: of an equilateral triangle where A = B = C = pie/3 radians so that each of a, b and c is equal to sqrt(3); next consider the case of a scalene triangle where A, B and C are all distinct. If A > pie/3, then B+C = pie-A = pie-(qty less than pie/3) and so either B or C is less than pie/3.
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Cheers
Vivek
Here's a question...
Let a, b, c be three non-zero real numbers such that
a + b + c = abc
Prove that at least one of these three numbers (a, b or c) is less than or equal to the square root of 3.
Can you prove this without trigonometry? The trigonometric solution follows...
Solution (using Trigonometry)
Let a = tan(A), b = tan(B), c = tan(C) (for some nonzero angles A, B, C which are real) so that the given constraint becomes
tan(A) + tan(B) + tan(C) = tan(A)tan(B)tan(C)
which can be true iff A + B + C = n*pie (n is an integer)
If n = 1, then A, B, C are the angles of a triangle (as the constraint is true for angles of a triangle). The result follows by considering cases: of an equilateral triangle where A = B = C = pie/3 radians so that each of a, b and c is equal to sqrt(3); next consider the case of a scalene triangle where A, B and C are all distinct. If A > pie/3, then B+C = pie-A = pie-(qty less than pie/3) and so either B or C is less than pie/3.
--------------------------------------------------------------------------
Cheers
Vivek